Public goods games:

Phase transitions

by Christoph Hauert, Version 2.2, September 2005.

Cooperators and defectors can co-exist only for a certain range of multiplication factors r. For low r cooperators vanish whereas for high r defectors are driven to extinction. When r approaches either one of the extiction thresholds, the system undergoes a critical phase transition. Numerical simulations suggest that the phase transition belongs to the universality class of directed percolation. Intuitively this is illustrated by noting that near the extinction thresholds small, isolated clusters of cooperators (defectors) meander around in a sea of defectors (cooperators). These clusters can divide, coalesce or annihilate which resembles branching and annihilating random walks. The phase transitions in these random walks are known to belong to the universality class of directed percolation (see references).

In the context of spatial games, a first-order phase transitions would be characterised by discontinous jumps in the strategy frequencies (in the limit of large population sizes) when changing r - but no such transitions were found to date. In contrast, second-order or critical phase transitions exhibit only a (non-differentiable) kink in the strategy frequency at the threshold value rx where at least one strategy disappears. More importantly, however, critical phase transitions are characterized by power-law scaling of various quantities when r approaches the threshold value. Minor changes in the microscopic updating procedure generally affect the threshold value but does not alter the exponents of the power-law scaling. Because of this robustness, systems are categorized in different universality classes according to their critical exponents.

The frequency fi of the strategy i decreases according to a power law: fi ∝ (r - rx)-β. In mean-field type transitions frequencies decrease linearly (β = -1). The directed percolation universality class in two dimensions is characterized by β = 0.58.

The spatial correlation length ξ ∝ (r - rx)-νdiverges according to another power law with a critical exponent ν.

According to the scaling hypothesis, near the critical threshold the correlation length ξ is the sole characteristic length scale in the system and all other quantities with dimensions of length must be measured in units of ξ. This assumption leads to simple scaling relations among the different critical exponents (see references).

The empirical support for critical phase transitions is based on extensive simulations that return the correct critical exponents for the decrease in frequency as well as for the diverging fluctuations of the vanishing strategy as well as for the diverging correlation lengths. Because of these diverging quantities, measurements require larger and larger system sizes when approaching the extinction threshold in order to avoid accidential extinctions. Estimations of suitable system sizes are obained by noting that the root-mean-square of the strategy fluctuations should be much smaller than their average frequency.

Critical phase transitions

The fraction of cooperators increases for increasing r. For r < 3 the benefits form cluster formation are insufficient and cooperators disappear. Conversely, for r > 3.65 defectors cannot persist and are displaced by cooperators. Considering the extinction of cooperators, the decrease in frequency seems consistent with a power-law. In contrast, the extinction of defectors looks more like a linear transition. However, there are indicators that the system exhibits another critical phase transition but that the power law would become visible only in the close vicinity of the extinction threshold. Detailed investigations of this transition are computationally intensive because of the exceedingly long relaxation times.

Click to enlarge

Critical phase transitions

The figure on the left shows the equilibrium frequency of cooperators (blue) and defectors (red) as a function of the multiplication factor r of the public good. Individuals are arranged on a square lattice with eight neighbors and interact in groups of size N = 5 with four randomly chosen neighbors. Each individual competes with a single randomly chosen neighbor and imitates the strategy of better performing neighbors with a probability proportional to the payoff difference (worse performing neighbors are not imitated).

The robustness of directed percolation transitions is nicely illustrated through comparisons of the critical phase transitions in compulsory public goods games and in voluntary public goods games, which denotes a variant where participation is optional. Individuals unwilling to participate in the public goods interaction are represented by a third strategic type, the loners. In the compulsory public goods game, the extinction of cooperators exhibits a directed percolation transition and so does the extinction of loners in the voluntary public goods game. However, the first transition leaves a homogenous state of defectors behind, whereas the extinction of loners occurs on an inhomogeneous and fluctuating background of cooperators and defectors.

While critical phase transitions are exiting for physicists, their presence alone may not be exceedingly important in biologically relevant scenarios. However, they do have substantial implications with far reaching consequences. For example, this demonstrates that small changes in one parameter can have tremendous effects on the equilibrium state of a system. But more importantly still, this indicates that in vulnerable systems it might be difficult, if not intrinsically impossible, to define characteristic scales in time and space that allow to fully understand the systems dynamics in empirical settings because both spatial and temporal correlation lengths diverge when approaching the critical threshold. This might be particularly relevant in conservation biology dealing with species interactions at the edge of extinction.

Scenarios

In order to investigate phase transitions it is best to adopt an updating mechanism inspired by condensed matter physics where an individual A adopts the strategy of B with the probability w = 1/(1 + exp[-z/T]) where z = PB - PA indicates the payoff difference between the two individuals and T represents some kind of temperature or noise term. At low temperatures, strategies of better performing individuals are adopted with high probabilities and those of worse performing individuals with small but non-zero probabilities. Conversely, at high temperatures the updating becomes random and quite indifferent to payoff differences. This update rule produces smooth relaxations and fast equilibration processes.

Legend

Time evolution of the frequency of cooperators in well-mixed populations where individuals engage in different kinds of interactions.

Near the extinction threshold of cooperators, cooperators form rather small compact clusters. The cluster movement resembles a branching and annihilating random walk. As r approaches the threshold value, the system exhibits a critical phase transition.

In analogy to the extinction of cooperators for low r, the system undergoes another critical phase transition at high r where cooperators drive defectors to extinction. Defectors form much smaller filament-like clusters but their movement again resembles a branching and annihilating random walk. Due to the smaller cluster size this analogy may be even more apparent in this case.